Optimal synchronization of oscillatory chemical reactions with complex pulse, square, and smooth waveforms signals maximizes Tsallis entropy

¹ The University of Electro-Communications - Tokyo 182-8585, Japan
² The Potsdam Institute for Climate Impact Research - Telegraphenberg A31, 14473 Potsdam, Germany
³ Department of Chemistry, Saint Louis University - St. Louis, MO 63103, USA

Received: 1 June 2015
Accepted: 24 August 2015

Abstract

We show that the mathematical structure of Tsallis entropy underlies an important and ubiquitous problem in nonlinear science related to an efficient synchronization of weakly forced nonlinear oscillators. The maximization of the locking range of oscillators with the use of phase models is analyzed with general constraints that encompass forcing waveform power, magnitude, or area. The optimization problem is then recasted as a general form of Tsallis entropy maximization. The solution of these optimization problems is shown to be a direct consequence from Hölder's inequality. The resulting new maximization principle is confirmed in numerical simulations and experiments with chemical oscillations with nickel electrodissolution. While weakly nonlinear oscillators have generic optimal waveforms (sinusoidal, 50% duty cycle square wave, and equally paced bipolar pulses for power-, area-, and magnitude-constraints, respectively), strongly nonlinear oscillators require more complex waveforms such as smooth, square, and pulse ones.